One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization ability and the modern deep learning paradigm of pretraining and finetuneing. However, this feature learning process remains poorly understood from a theoretical perspective, with existing analyses largely restricted to two-layer networks. In this work we show that three-layer neural networks have provably richer feature learning capabilities than two-layer networks. We analyze the features learned by a three-layer network trained with layer-wise gradient descent, and present a general purpose theorem which upper bounds the sample complexity and width needed to achieve low test error when the target has specific hierarchical structure. We instantiate our framework in specific statistical learning settings -- single-index models and functions of quadratic features -- and show that in the latter setting three-layer networks obtain a sample complexity improvement over all existing guarantees for two-layer networks. Crucially, this sample complexity improvement relies on the ability of three-layer networks to efficiently learn features. We then establish a concrete optimization-based depth separation by constructing a function which is efficiently learnable via gradient descent on a three-layer network, yet cannot be learned efficiently by a two-layer network. Our work makes progress towards understanding the provable benefit of three-layer neural networks over two-layer networks in the feature learning regime.

Biological and artificial learners are inherently exposed to a stream of data and experience throughout their lifetimes and must constantly adapt to, learn from, or selectively ignore the ongoing input. Recent findings reveal that, even when the performance remains stable, the underlying neural representations can change gradually over time, a phenomenon known as representational drift. Studying the different sources of data and noise that may contribute to drift is essential for understanding lifelong learning in neural systems. However, a systematic study of drift across architectures and learning rules, and the connection to task, are missing. Here, in an online learning setup, we characterize drift as a function of data distribution, and specifically show that the learning noise induced by task-irrelevant stimuli, which the agent learns to ignore in a given context, can create long-term drift in the representation of task-relevant stimuli. Using theory and simulations, we demonstrate this phenomenon both in Hebbian-based learning -- Oja's rule and Similarity Matching -- and in stochastic gradient descent applied to autoencoders and a supervised two-layer network. We consistently observe that the drift rate increases with the variance and the dimension of the data in the task-irrelevant subspace. We further show that this yields different qualitative predictions for the geometry and dimension-dependency of drift than those arising from Gaussian synaptic noise. Overall, our study links the structure of stimuli, task, and learning rule to representational drift and could pave the way for using drift as a signal for uncovering underlying computation in the brain.

If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps.

Neural networks are vulnerable to adversarial examples, i.e. inputs that are imperceptibly perturbed from natural data and yet incorrectly classified by the network.